ESBSD: AN ESSAY ON THE NEW EXPONENTIAL SMOOTHING METHODOLOGY
APPLIED TO THE PROJECTION OF THE POPULATION OF BELO HORIZONTE
Daniel Henrique Pereira¹
¹ Student, Business Administration, Pontifical Catholic University of Minas Gerais (PUC Minas)
Abstract: Population projection is essential for the public and private sectors, because it is possible
to estimate the number of people that will be served by infrastructure works and services. In this
paper, a mathematical model based on alternative exponential smoothing was presented, tested and
approved in thousands of real case studies in different fields of study. Using this method and after
obtaining data from previous years, projections were raised for the years 2022 to 2034 for the city
of Belo Horizonte, Minas Gerais, Brazil.
Keywords: Population Projection; Exponential Smoothing; Time Series; Statistics; Belo Horizonte
According to Abubakar and Usaini (2019) population projection is essential for public and private
policy planning, as it helps to make decisions about the resources needed to meet the demand of a
specific geographical location. When it comes to the public sector, population projection, when
done well, will meet the needs of the population through infrastructure works, avoiding unnecessary
costs with projections overestimating actual values and projections underestimating actual
population values that imply the fact a certain majority of the population will not have their needs
fully met. Therefore, we must find a "balance" a method that minimizes errors.
Generally, when we discuss population projections, we also turn to a concept called Time Series. As
Adhikari and Agrawal (2013) say, time series is a dynamic field of study in which its main goal, is
to collect data from past observations and through them identify patterns that can be useful for
making future predictions.
According to Gawatre et al. (2016) we have different methodologies and each of them can be useful
at certain stages of population dynamics. In this sense, this paper aims to present an alternative
mathematical model for projecting population growth in the city of Belo Horizonte, Brazil. Through
data obtained by PopulationStat, projections for the years 2022 to 2034 will be raised.
II MATERIALS AND METHODS
The author collected data from 2017 to 2021 from the Belo Horizonte population present in the data
provided by the PopulationStat platform and also data on the future population of Belo Horizonte
obtained through projections registered by the same platform. As a quantitative measure of the
prediction error, the Absolute and Relative Error was considered.
III ESBSD METHODOLOGY
3.1 Population of the city of Belo Horizonte between 2017 to 2021
Historical series of the population of Belo Horizonte for the years 2017 to 2021 was according to
the data collected through the PopulationStat platform that collects its data from official sources of
each country and other data such as World Bank and United Nations.
Population of the city of Belo Horizonte between 2017 and 2021, data by PopulationStat
Time Years Population Growth Rate (±Δ%)
1 2017 5,899,000 -
2 2018 5,972,000 +1.24%
3 2019 6,028,000 +.94%
4 2020 6,084,000 +.93%
5 2021 6,140,000 +.92%
3.2 Understanding ESBSD Methodology
In 2018, I have been writing for the first time a draft about the methodology will be presented ahead
due to some content I had as a business student. Only now has it been possible to present it. The
ESBSD model, which stands for "Exponential Smoothing Based on Standard Deviation", is a
forecasting model applied to environments in which the observed variable has a constant growth or
decay and with similar behavior as Delmas (2004) states about the so-called Saturation Population
concept was introduced by Pierre François Verhulst through his proposal with Logistic Regression.
Figure 1. Logistic Regression and the concept of Saturation Population by Pierre François Verhulst
The motivation behind the presentation of this model is the fact in certain sets of data present in a
time series with characteristics of constant growth or decay trend, even choosing the lowest or the
highest value of the constants, usually lies between 0 and 1, it is still not possible for the future
projections Ft+n to follow real values. In this sense, ESBSD presents itself as a way to minimize
these issues by "widening" a range of possible future outcomes.
As an introduction, this model can be represented by the equation below:
Ft+1 = Future value
γ = Weighting factor between 0 and 1
At-1 = Last analyzed period
n = “n” past periods
Using this method, we should note that on one side of the formula there is the last period t-1
multiplied by an "arbitrary" weighting factor between 0 and 1, and on the other side, there is the
multiplication of this same weighting factor by the square root of the "basic growth/decline rate"
which is given by the ratio between the last period analyzed with the average of "n" periods of past
observations and then multiplying again by the last period we find a relative value of what may
occur in the future.
Besides what was previously mentioned, in the ESBSD there is also a "Complementary Factor ±"
(CF±) in which the result found will be added or subtracted from the results obtained through the
calculation performed by the CF±.
Through this CF± present in the formula is derived its name (ESBSD), since, in a historical series if
the values of the n periods observed are dispersed, we will have a high standard deviation,
consequently when we subtract by the difference between the mean and each of its component
values, we will also have a relatively high CF± value. Similarly, if we take into account a historical
series in which data are less scattered around the mean, then the CF± will also have a relatively
lower value. Clear and understandable the behavior of the standard deviation of an analyzed data set
together with the weighting factor, strongly influences the future results we want to project.
In summary, if by historical series we have an environment with constant growth, it will be
considered CF+ (a positive value) in a similar way if we have an environment with constant decay
we will soon consider a CF- (a negative value).
Below is the CF± formula:
This CF± consists of subtracting the same standard deviation from n time periods by the difference
between the mean and each of its component values from this same n periods considered. Absolute
Value Function (ABS) in this case is intended to balance the data in the series, i.e. preventing a
possible outlier in the growth/decay rate in an n time period from significantly distorting the data,
making a good future projection impossible.
Therefore, an average of each of these "remaining" values must be taken until the value of the CF±
It is interesting to share if the data of a given time series presents a high level of volatility and/or
regardless of the smallest or largest constant value is not able to approximate the actual values in
the forecast simulations, it is interesting to disregard the use of the square root over the ratio
between the last analyzed period with the average of n considered periods, as we can see below:
In this way, it will be possible to expand the range of possibilities so that we have more freedom
with the choice of constants and that they can be useful in the medium and long term and above all,
they can approximate actual values in the simulations and in the future to support decision making.
Through thousands of case studies analyzed from business administration (sales forecasting) to
demography and biology, the use of the square root has demonstrated recurrent practical
applicability. However, you will have some cases where without its use it can be more interesting as
will be ahead presented. In this sense, as in any other similar method, it is recommended that the
user previously perform tests to identify which will be the best value of the constant between 0 and
1 that approximates the actual values.
After emphasizing some particularities that should be considered, below we can see the general
formula that contains CF±:
3.3 ESBSD Calculation Process
The steps to reach the final result of the ESBSD projection consist of the following steps:
1st: Define the number of n periods to be considered by the "basic rate of growth/decay"
with respect to the average considered (3 n periods are recommended in this study).
2nd: Define the number of n periods to be considered in the "complementary factor ±" (4 n
periods are recommended in this study).
3rd: Define a value between 0 and 1 for the "weighting factor".
Taking into account the stages of the ESBSD, the "projection" for the year 2021 and a comparison
of the actual result obtained by the PopulationStat of the same year will be carried out, in order to
evaluate its practical applicability.
3.3.1 Population projection for 2021
1°: Number of n periods to be considered by the "basic growth/decay rate" will be n = 3;
2°: Number of n periods to be considered by the "Complementary Factor ±" will be n=4;
3°: "Weighting Factor" to be considered will be γ = 0.538
The same specifications will apply for the projection of the following years.
Considering the data presented in table 1:
= 5,995,750.00 S4 = 79,062.74
ABS [79,062.74 – ABS (5,995,750.00 – 6,084,000)] = 9,187.26
ABS [79,062.74 – ABS (5,995,750.00 – 6,028,000)] = 46,812.74
ABS [79,062.74 – ABS (5,995,750.00 – 5,972,000)] = 55,312.74
ABS [79,062.74 – ABS (5,995,750.00 – 5,899,000)] = 17,687.26
We can notice in the process of calculating CF± we can find negative values both in the subtraction
between the average and each of the component values of the set of n past periods, as in the
"remaining" values, but in this case, we will consider the same values found, but with a positive
sign, because we are considering the Absolute Value Function ABS that, according to Tao (2011)
among its properties is non-negativity | f | : x → | f (x)| and | x | = 0 if an only if x = 0.
Another point to note in this case study is that since n number ≠ 0 of this time series is ∀ ∈ ℝ
shown to grow continuously despite the growth rate tending to 0 (zero) over time we can ≅ ∴
consider a positive sign for CF±.
CF± 2021 = 9,187.26 + 46,812.74 + 55,312.74 + 17,687.26 / 4
CF± 2021= + 32,250.00
After performing the calculation of CF± which will be used in the formula, the calculation will be
performed on the second side of the model where an average of the last 3 periods was considered:
We can note in this case study, the particularity of the fact occurred in our first test (exclusive for
the choice of the value of the constant considered ideal) that regardless of using the smallest or
largest value of the constant between 0 and 1 it was still not possible to reach a range in which it
approximates the actual values (2021). For this reason, it was used in this second part of the
formula, the ratio between the last analyzed period with the average of n time periods without
extracting the square root. As the same tends to happen with other similar exponential smoothing
models, this example reinforces the main reason for proposing this model.
Using the second side of the model:
Therefore, we get as value = 6,140,520.24.
Final calculation of the model is then reached:
Ft 2021 = 0,538 (6,084,000) + 0,462 (6,140,520.24) + 32,250.00
Ft2021 = 3,273,192.00 + 2,836,920.35 + 32,250.00
Ft2021 = 6,142,362
Therefore, 6,142,362 would be the number of inhabitants for the city of Belo Horizonte according
to the ESBSD model.
As Makridakis et al. (1983) states, the Absolute and Relative (%) Error analysis is very useful in
time series analysis. In this sense, below - in table 2 - is the absolute and relative error for the model
taking into account the comparison with the actual data obtained by the PopulationStat in 2021:
Comparison of the results of the ESBSD model with actual data obtained by the PopulationStat
Population by ESBSD
(2021) Absolute Error Relative Error
6,140,000 6,142,362 2,362 .038%
In the same sense, the population will be projected for the subsequent years from 2022 to 2034.
3.3.2 Population projection for 2022
= 6,056,000.00 S4 = 72,295.69
ABS [72,295.69 – ABS (6,056,000.00 – 6,140,000)] = 11,704.31
ABS [72,295.69 – ABS (6,056,000.00 – 6,084,000)] = 44,295.69
ABS [72,295.69 – ABS (6,056,000.00 – 6,028,000)] = 44,295.69
ABS [72,295.69 – ABS (6,056,000.00 – 5,972,000)] = 11,704.31
It should be noted that for the 2022 projection the last actual value (6,140,000) was considered part
of the n=4 elements, carried out by the 2021 census and not the ESBSD projection in the same year,
because we still have actual data. Below, the calculations continue:
CF± 2022 = 11,704.31 + 44,295.69 + 44,295.69 + 11,704.31 / 4
CF± 2022= + 28,000.00
Ft 2022 = 0,538 (6,140,000) + 0,462 (6,196,515.45) + 28,000.00
Ft2022 = 3,303,320.00 + 2,862,790.14 + 28,000.00
Ft2022 = 6,194,110
3.3.3 Population projection for 2023
= 6,111,527,50 S4 = 71,566.19
ABS [71,566.19 – ABS (6,111,527,50 – 6,194,110] = 11,016.31
ABS [71,566.19 – ABS (6,111,527,50 – 6,140,000] = 43,093.69
ABS [71,566.19 – ABS (6,111,527,50– 6,084,000)] = 44,038.69
ABS [71,566.19 – ABS (6,111,527,50 – 6,028,000)] = 11,961.31
In the 2023 projection, the last value found (2022) will be considered part of the n=4 elements, but
as we do not have actual values available, the projection given by the ESBSD in the year 2022 will
be considered as "actual value". This same reasoning will be considered for the other years.
CF± 2023 = 11,016.31 + 43,093.69 + 44,038.69 + 11,961.31 / 4
CF± 2023= + 27,527.50
Ft 2023 = 0,538 (6,194,110) + 0,462 (6,249,338.07) + 27,527.50
Ft2023 = 3,332,431.18 + 2,887,194.19 + 27,527.50
Ft2023 = 6,247,153
3.3.4 Population projection for 2024
= 6,166,315.75 S4 = 70,179.73
ABS [70,179.73 – ABS (6,166,315.75 – 6,247,153)] = 10,657.52
ABS [70,179.73 – ABS (6,166,315.75 – 6,194,110)] = 42,385.48
ABS [70,179.73 – ABS (6,166,315.75 – 6,140,000)] = 43,863.48
ABS [70,179.73– ABS (6,166,315.75– 6,084,000)] = 12,136.52
CF± 2024 = 10,657.52 + 42,385.48 + 43,863.48 + 12,136.52 / 4
CF± 2024= + 27,260.75
Ft 2024 = 0,538 (6,247,153) + 0,462 (6,301,012.04) + 27,260.75
Ft2024 = 3,360,968.31 + 2,911,067.56 + 27,260.75
Ft2024 = 6,299,297
3.3.5 Population projection for 2025
= 6,220,140.00 S4 = 68,545.64
ABS [68,545.64 – ABS (6,220,140.00– 6,299,297)] = 10,611.36
ABS [68,545.64 – ABS (6,220,140.00 – 6,247,153)] = 41,532.64
ABS [68,545.64 – ABS (6,220,140.00 – 6,194,110)] = 42,515.64
ABS [68,545.64 – ABS (6,220,140.00 – 6,140,000)] = 11,594.36
CF± 2025 = 10,611.36 + 41,532.64 + 42,515.64 + 11,594.36 / 4
CF± 2025= + 26,563.50
Ft 2025 = 0,538 (6,299,297) + 0,462 (6,352,180.94) + 26,563.50
Ft2025 = 3,389,021.79 + 2,934,707.59 + 26,563.50
Ft2025 = 6,350,293
Using the same reasoning, we proceed with the calculation of the population projection until 2034.
The results found through the ESBSD method for the years 2024 to 2039 were as follows according
to the table below:
Belo Horizonte’s population for the years 2022 to 2034 according to the ESBSD methodology
Years Population by ESBSD
Methodology Growth Rate (%)±
2022 6,194,110 + .88%
2023 6,247,153 + .86%
2024 6,299,297 + .83%
2025 6,350,293 + .81%
2026 6,400,233 + .79%
2027 6,449,173 + .76%
2028 6,497,097 + .74%
2029 6,544,027 + .72%
2030 6,589,988 + .70%
2031 6,634,994 + .68%
2032 6,679,062 + .66%
2033 6,722,210 + .65%
2034 6,764,455 + .63%
Below, in Table 4, the proposed methodology will be compared with the projections for the
following years presented by the PopulationStat platform, which takes into consideration too many
other variables in its calculation, such as national wealth, quality of life index, birth rate, fecundity
and mortality, for example.
A comparison between the projection made by the ESBSD methodology and that made by the
Years Population by
Population by ESBSD
2022 6,194,000 6,194,110 + 110
2023 6,248,000 6,247,153 - 847
2024 6,300,000 6,299,297 - 703
2025 6,352,000 6,350,293 - 1,707
2026 6,402,000 6,400,233 - 1,767
2027 6,450,000 6,449,173 - 827
2028 6,496,000 6,497,097 + 1,097
2029 6,541,000 6,544,027 + 3,027
2030 6,583,000 6,589,988 + 6,988
2031 6,624,000 6,634,994 + 10,994
2032 6,663,000 6,679,062 + 16,062
2033 6,699,000 6,722,210 + 23,210
2034 6,734,000 6,764,455 + 30,455
As a complementary information and of importance for the following projections, Instituto
Brasileiro de Geografia e Estatística (Brazilian Institute of Geography and Statistics, IBGE) through
its studies with multiple variables strongly considers the possibility the population of Belo
Horizonte, as well as the whole of Brazil, will decline after 2040, in this sense, it is sensible and
realistic to make projections with mathematical models related to the growth until the given period.
It is important to note that, as in other exponential smoothing models, in ESBSD, the higher the
value of the chosen constant, means that the user will be giving more importance to past data,
likewise, the lower the value of the constant, the more importance will be given to more recent data.
Another issue to highlight is that the model is adaptable with the input of new data, thus adapting to
new changes in the growth/decay dynamics.
Through this case study applied to population projection, ESBSD has again demonstrated
interesting applicability. Despite being a mathematical model, it was able to adapt to the trends
expected by official agencies such as IBGE (Instituto Brasileiro de Geografia e Estatística) and the
United Nations (whose data is presented by the PopulationStat platform) as to the future number of
inhabitants expected for the population of the city of Belo Horizonte by the year 2034.
Through table 4, for example, we can see the sum of the values between the years 2022 to 2034,
when compared with the values presented by PopulationStat, returns a total difference of 86,092.00.
With this result we can affirm its potential high level of accuracy for the following years if the
projections also raised by official agencies such as IBGE and the United Nations are on the same
path of the actual values in which we will know in the future.
Population projection, when properly carried out may contribute to the planning of public policies
and the private sector and consequently contribute to the economic and social development of a
given geographic location under analysis, since resources will tend to be applied in a more rational
manner while seeking to meet the needs of the population and avoiding the lack and misuse of
ESBSD, in its many experiments considering real case studies (from sales forecasting to areas such
as demographics and biology, for example) and a continuously growing/decreasing trending
environment (additionally, also trending towards a "Saturation Population" as Pierre François
Verhulst raised in his study), can present a practical applicability in the short, medium and long term
if the value of the constant between 0 and 1 is well established by the user in his testing and
On the other hand, as in any other similar model, among its possible limitations, we can mention the
fact that its application is restricted to only certain types of environments as mentioned in this paper.
Moreover, its calculation process may contain “additional steps” than normally expected, but in
general it is compensated by the high level of accuracy obtained according to the means of
Considering the case study with the survey of the population growth projection of the city of Belo
Horizonte, it was possible to observe its practical applicability only in the short term, because
official agencies of Statistics of Brazil (IBGE) points to a strong trend of population decay of the
Brazilian population from 2040, when taking into account too many other variables intrinsic to the
field of study of demography as birth rate; fertility, mortality, in addition to other important
indicators such as national wealth, migration; emigration; quality of life index, unemployment rate,
access to education, among other examples directly influence the population dynamics.
Mathematical models, when applied to population projections, can be useful in aiding decision-
making when a given geographic location does not have a complete Demographic Census and has
too many resource constraints. With regard to ESBSD, whether in population projection analysis or
in other fields of study, it has proven to be an interesting model for its proposal. In this sense, it
remains as one more model for the portfolio to successfully address real problems.
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